Problem 15 - Thomas Calculus

Question

Which of the series in Exercises \(15 - 48\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } ( 0.1 ) ^ { n } $$

Step-by-Step Solution

Answer

The series converges absolutely.

Step 1: Understand the Series

The series in question is given by \( \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } ( 0.1 ) ^ { n } \). This is an alternating series due to the \(( - 1 ) ^ { n + 1 }\) factor.

Step 2: Check the Absolute Convergence

To check if the series converges absolutely, we consider the series without the alternating signs: \( \sum _ { n = 1 } ^ { \infty } ( 0.1 ) ^ { n } \). This is a geometric series with the common ratio \( r = 0.1 \). Since \(|r| < 1\), the series converges absolutely.

Step 3: Check for Convergence

The original series is an alternating series. To check for convergence, consider the Alternating Series Test: The terms \((0.1)^{n}\) are positive, decreasing, and approach zero as \( n \to \infty \). Thus, the series converges.

Step 4: Conclusion

Since the series converges absolutely, and also converges according to the Alternating Series Test, the series is both absolutely convergent and convergent.

Key Concepts

Alternating Series

An alternating series is a series where the terms alternate in sign. This means that the series is of the form

• For example, \[ \sum _{n=1}^{\infty} (-1)^{n+1} a_n \]

with:

• An example of an alternating series was given by the series:\[ \sum _{n=1}^{\infty} (-1)^{n+1} (0.1)^n \]

The alternating part implies a change in signs between terms. So it's like adding a number in one step and then subtracting a number in the next. This can often help the series converge.

Geometric Series

A geometric series is a series where each term is a constant multiple of the previous term. It takes the form:

• \[ \sum_{n=0}^{\infty} ar^n \]

Here, "a" is the first term, and "r" is the common ratio. If you can identify this pattern, you're looking at a geometric series.We determine convergence based on the absolute value of the common ratio:

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

In our case, after removing the alternating sign, the geometric series \( \sum _{n=1}^{\infty} (0.1)^n \)has \(r = 0.1\). Since \(|0.1| < 1\), the series converges.

Absolute Convergence

Absolute convergence is a specific type of convergence in series. If a series converges even when all its terms are replaced by their absolute values, then it converges absolutely.

• For example, consider the series: \[ \sum _{n=1}^{\infty} |(-1)^{n+1} (0.1)^n| = \sum _{n=1}^{\infty} (0.1)^n \]

If \( \sum _{n=1}^{\infty} a_n \) converges, and \( \sum _{n=1}^{\infty} |a_n| \) also converges, then the series has absolute convergence.This is a stronger form of convergence because it indicates the series behaves well even without considering alternating signs.

Alternating Series Test

The Alternating Series Test is a helpful tool to check the convergence of alternating series. To apply it, follow these conditions:

• The absolute value of the terms \((a_n)\) must decrease steadily.
• The limit of \(a_n\) as \(n \to \infty\) is zero.

If both conditions are met, the alternating series converges.For example, in the exercise series\[ \sum _{n=1}^{\infty} (-1)^{n+1} (0.1)^n \], since the terms \((0.1)^n\) meet these criteria, the series converges by the Alternating Series Test.